MontréalMars 2001

Série ScientifiqueScientific Series

2001s-20

Estimating the IntergenerationalEducation Correlation from aDynamic Programming Model

Christian Belzil, Jörgen Hansen

CIRANO

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Les organisations-partenaires / The Partner Organizations

•École des Hautes Études Commerciales•École Polytechnique•Université Concordia•Université de Montréal•Université du Québec à Montréal•Université Laval•Université McGill•MEQ•MRST•Alcan inc.•AXA Canada•Banque du Canada•Banque Laurentienne du Canada•Banque Nationale du Canada•Banque Royale du Canada•Bell Québec•Bombardier•Bourse de Montréal•Développement des ressources humaines Canada (DRHC)•Fédération des caisses populaires Desjardins de Montréal et de l’Ouest-du-Québec•Hydro-Québec•Imasco•Industrie Canada•Pratt & Whitney Canada Inc.•Raymond Chabot Grant Thornton•Ville de Montréal

© 2001 Christian Belzil et Jörgen Hansen. Tous droits réservés. All rights reserved.Reproduction partielle permise avec citation du document source, incluant la notice ©.Short sections may be quoted without explicit permission, if full credit, including © notice, is given to the source.

ISSN 1198-8177

Ce document est publié dans l’intention de rendre accessibles les résultats préliminairesde la recherche effectuée au CIRANO, afin de susciter des échanges et des suggestions.Les idées et les opinions émises sont sous l’unique responsabilité des auteurs, et nereprésentent pas nécessairement les positions du CIRANO ou de ses partenaires.This paper presents preliminary research carried out at CIRANO and aims atencouraging discussion and comment. The observations and viewpoints expressed are thesole responsibility of the authors. They do not necessarily represent positions of CIRANOor its partners.

Estimating the Intergenerational Education Correlationfrom a Dynamic Programming Model*

Christian Belzil†, Jörgen Hansen‡

Résumé / Abstract

À l'aide d'un modèle de programmation dynamique, nous analysonsl'importance relative des antécédents familiaux et de l'habilité non-observée dansla détermination du niveau de scolarité et des salaires. Nous évaluons aussi lacorrélation intergénérationelle au niveau de l'éducation et déterminons l'effet d'uneaugmentation exogène du niveau de scolarité de la génération présente sur leniveau d'éducation de la génération future.

Using a structural dynamic programming model, we investigate therelative importance of initial household human capital endowments andunobserved individual abilities in explaining cross-sectional differences inschooling attainments and wages. We evaluate the true intergenerationaleducation correlation and the effect of an exogenous increase in human capital ofthe current generation on schooling attainments of the next generation. We findthat the variation in schooling attainments explained by differences in householdhuman capital is twice as large as the variation explained by unobserved abilities.However, the variation in labor market wages explained by differences inunobserved abilities is 3 times larger than the variation explained by householdhuman capital endowments. We also find that the true partial correlationsbetween son and father’s schooling and between son and mother’s education arerespectively 17% and 40% lower than the sample correlations. Increasing thelevel of schooling of the current generation by one year raises schoolingattainments of the next generation by 0.4 year.

Mots Clés : Corrélation intergénérationelle de l'éducation, rendements en éducation, capitalhumain du foyer, croissance, décision d'études

Keywords: Intergenerational education correlation, returns to education, household humancapital, growth, schooling decision

JEL: J2, J3

* Corresponding Author: Christian Belzil, CIRANO, 2020 University Street, 25th floor, Montréal, Qc, CanadaH3A 2A5 Tel.: (514) 985-4000 Fax: (514) 985-4039 email: belzilc@cirano.qc.caAn earlier version of this paper was presented at the Conference "The Econometrics of Strategic Decision Making''which was held at Yale University (Cowles Foundation) in May 2000. We would like to thank Jean-Marc Robin andJohn Rust for useful comments. Belzil thanks the Social Sciences and Humanities Research Council of Canada forgenerous funding. The usual disclaimer applies.† Concordia University and CIRANO‡ Concordia University

1 Introduction

Individual schooling attainments are one of the key components of the level ofhuman capital in an economy. They are an important determinant of incomedistribution and are often thought to be one of the key factors explaining thewealth of nations as well as cross-nation di�erences in economic growth. Indeed,the recent revival of neo-classical growth models is largely based on human capitaltheory (Lucas, 1988, Barro and Sala-i-Martin, 1995).1

At the micro level, it is customary to assume a strong correlation betweenone's schooling attainment and parents' (or household) education. The e�ects ofhousehold background variables on individual schooling attainments can take var-ious forms. While enrolled in school, young individuals typically receive parentalsupport. Although parental support is usually unobservable to the econometri-cian, it is expected to be highly correlated with household human capital andincome. At the same time, innate ability, also correlated with household humancapital, should have an impact on the decision to attend school and on labormarket wages.

The net e�ects of household human capital on individual schooling attain-ments are far from obvious. On the one hand, households who have higher in-come may transfer more resources to their children and reduce substantially theopportunity cost of school attendance. On the other hand, wealthier householdsalso face a higher opportunity cost of spending time with children and may reducetheir investment in children. The e�ect of innate ability on school attendanceis also unclear. If skill endowments are strongly correlated with household hu-man capital (especially father's and mother's education), those young individualsraised in households endowed with a high level of human capital will have a highlevel of school ability but will also have a high level of market ability (absoluteadvantage in the labor market).

Whether individual schooling attainments are more a�ected by household hu-man capital or by innate ability remains an open question. Nevertheless, laboreconomists should be skeptical of drawing strong conclusions on the causal ef-fect of parents' human capital from the empirical correlations between varioushousehold background variables and individual schooling attainments, which areobtained from OLS regressions. For instance, reduced-form OLS estimates of thee�ects of household human capital on schooling attainments ignore dynamic di-mensions and cannot distinguish between the utility of attending school and labormarket outcomes. They are also incapable of identifying the relative importanceof individual ability and pure stochastic shocks.

The e�ect of parental background on educational achievements has been welldocumented in a reduced-form framework (Kane, 1994 and Lazear, 1980), as well

1Although the links between schooling and private wages is well established at the microlevel, the relationship between economic growth and education is substantially weaker. Thisparadox is currently the object of a large amount of work (see Topel, 1999, for a survey).

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as in a semi-structural framework (Cameron and Heckman, 1998 and Magnacand Thesmar, 1998). As of now, the e�ects of household background on schoolingattainments have only rarely been investigated within a full structural framework.Eckstein and Wolpin (1999) have estimated a �nite mixture model of schoolattendance and work behavior. While their model does not allow them to estimatethe direct e�ect of household background variables, they can merge actual data onschooling attainments with data on household characteristics and use Bayes' ruleto relate those data to unobserved type probabilities. Belzil and Hansen (2001)use a dynamic programming model of schooling decisions in order to estimatethe returns to schooling. In this model, the utility of attending school dependsexplicitly on household background variables. In both of these papers there isevidence that school attendance increases with household human capital althoughthe relative importance of household characteristics and unobserved abilities arediÆcult to evaluate.

Although the notion of a \true" intergenerational education correlation hasnot raised much interest amongst empirical labor economists, it naturally arisesin the dynamic macroeconomic literature concerned with economic growth, over-lapping generations and intergenerational transfers. If the true intergenerationaleducation correlation can be inferred from micro data, it can allow economiststo simulate the e�ect of an exogenous increase in human capital (accompaniedby the resulting growth in income) of the current generation on schooling at-tainments and labor market productivity of the next generation. This may helpdetermine if education is a consequence as well as a cause of economic growth.

The main objective of the present paper is to estimate a structural model ofschooling decisions in which the separate e�ects of unobserved abilities and house-hold human capital endowments on the key determinants of schooling attainments(the instantaneous utility of attending school and labor market outcomes) can beidenti�ed. We pay particular attention to the following 4 questions.

1. How much of individual di�erences in schooling attainments is explainedby individual heterogeneity in unobserved school and market abilities asopposed to di�erences in household human capital endowments (father'seducation, mother's education, household income and the like)?

2. How much of predicted wages is explained by individual heterogeneity inschool and market abilities as opposed to di�erences in household humancapital endowments?

3. Does the intergenerational education correlation, predicted by the struc-tural dynamic programming model, di�er substantially from the samplecorrelation?

4. What is the e�ect of an exogenous increase in the level of education andincome of the current generation on schooling attainments of the next gen-eration?

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As far as we know, none of these questions have been answered to date. Fol-lowing Belzil and Hansen (2001), we estimate a �nite horizon dynamic program-ming model which is solved using recursive methods. In our model, schoolingdecisions a�ect future wages and lifetime employment rates. Individuals are en-dowed with exogenous household characteristics and innate abilities. Both ofthese a�ect the utility of attending school and labor market outcomes. Individ-uals share a common rate of time preference. Labor market ability a�ects bothwages and employment rates. We assume that individual ability is the sum ofa deterministic (observable) component capturing the e�ect of household humancapital and a stochastic (unobserved) component representing idiosyncratic abil-ity which is orthogonal to household human capital. Given their endowments,individuals decide on the optimal allocation of time between school attendanceand the labor market.

The estimation of our model is computer intensive. In order to estimate amodel where the degree of exibility is high enough to capture the importanceof both unobserved ability and observed initial endowments in household humancapital, we assume that population unobserved abilities can be described by 6discrete types of individuals. The model must therefore be solved recursively 6times for each individual. For this reason, we concentrate on a model speci�cationwhich can be solved in closed-form. The model is implemented on a panel ofwhite males taken from the National Longitudinal Survey of Youth (NLSY). Thepanel covers the period from 1979 until 1990.

The results show that there is overwhelming evidence that household humancapital variables a�ect both the utility of attending school and labor marketoutcomes. Our estimates indicate that the di�erences in predicted schoolingattainments explained by household human capital are generally twice as largeas the di�erences explained by unobserved abilities. On the other hand, thedi�erences in predicted wages explained by unobserved abilities are three timesas large as the di�erences explained by household human capital. These resultsare easily explained. First, household human capital has a much larger e�ecton the utility of attending school than on labor market outcomes. Second, thewage return to schooling is found to be quite low so that individual di�erencesin schooling do not explain di�erences in wages accurately. Third, labor marketability appears to be the prime factor explaining predicted wages.

We �nd that the true intergenerational education correlation predicted bythe dynamic programming model is much lower than the correlation obtainedby standard OLS regressions of observed schooling on household human capitalvariables. The true partial correlation between son and father's schooling (around0.17) is 17% lower than the correlation found in the data (around 0.21). The truepartial correlation between son and mother's education (around 0.11) is 40% lowerthan correlation disclosed by the data (0.17). The structural model also predictsthat the true correlation between schooling attainments and household incomeand the correlation between schooling attainments and the number of siblings are

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lower than the sample correlation.Finally, these estimates imply that an exogenous increase of 1 year in the level

of schooling of the current generation will increase schooling attainments of thenext generation by 0.4 year.

The main features of the paper are the following. Section 2 is devoted tothe presentation of the dynamic programming model. Section 3 contains a briefdescription of the sample used in this paper (NLSY). The main empirical resultsare discussed in Section 4. The conclusion is in Section 5.

2 The Model

Individuals are initially endowed with household human capital, innate abilityand a rate of time preference (denoted �). Given their endowments, young indi-viduals decide sequentially whether it is optimal or not to enter the labor marketor continue accumulate human capital. Individuals maximize discounted expect-ed lifetime utility. The control variable, dt; summarizes the stopping rule. Whendt = 1; an individual invests in an additional year of schooling at the beginningof period t. When dt = 0, an individual leaves school at the beginning of periodt (to enter the labor market). Every decision is made at the beginning the periodand the amount of schooling acquired by the beginning of date t is denoted St:As it is diÆcult to write down a full structural model which would include allthe e�ects that household human capital variables may have on the probabilityof transiting from one grade level to the next, we specify a reduced-form functionfor the utility of attending school. The function is allowed to depend on varioushousehold background variables as well as individual unobserved ability.

The instantaneous utility of attending school is

U school(:) = X 0

iÆ + (Sit) + ��i + "

�it (1)

where Xi contains the following variables: father's education, mother's educa-tion, household income, number of siblings, household composition at age 14 andregional controls. The number of siblings is used to control for the fact that,other things equal, the amount of parental resources spent per child decreaseswith the number of siblings. The household composition variable (Nuclear Fam-ily) is equal to 1 for those who lived with both their biological parents (at age14) and is likely to be correlated with the psychic costs of attending school. Thegeographical variables are introduced in order to control for the possibility thatdirect (as well as psychic) costs of schooling may di�er between those raised inurban areas and those raised in rural areas, and between those raised in the southand those raised in the north. Yearly household income is measured in units of$1,000. The term �

�i represents individual heterogeneity (ability) a�ecting the

utility of attending school. It is discussed in more details below. The utilityof attending school is allowed to depend on the level of schooling in a exible

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fashion. This is done using a spline function approximation of (St). Finally,"�t represents a stochastic utility shock and is assumed to be i:i:d Normal withmean 0 and variance �2� :

We assume that individuals interrupt schooling with exogenous probability�(St) and, as a consequence, the possibility to take a decision depends on a statevariable It: When It = 1; the decision problem is frozen for one period. If It = 0;the decision can be made. The interruption state is meant to capture events suchas illness, injury, travel, temporary work, incarceration or academic failure. Whenan interruption occurs, the stock of human capital remains constant over theperiod. The NLSY does not contain data on parental transfers and, in particular,does not allow a distinction in income received according to the interruptionstatus. As a consequence, we ignore the distinction between income support atschool and income support when school is interrupted.2

Once the individual has entered the labor market, he receives monetary in-come ~wt; which is the product of the yearly employment rate, et; and the wagerate, wt: The instantaneous utility of work

Uwork(:) = log( ~wt) = log(et � wt)

The log wage received by individual i, at time t, is given by

logwit = '1(Sit) + '2:Experit + '3:Exper2it + �wi + "wit (2)

where '1(St) is the function representing the wage return to schooling. Both '2

and '3 are parameters to be estimated and �wi is unobserved labor market abilitya�ecting wages.

To characterize the stochastic process of the employment security variable, et;we assume that

log(e�it) = �it + "eit

where e�it = log( 1eti) and where "eit is a random shock normally distributed with

mean 0 and variance �2e :3 The employment rate is also allowed to depend on

accumulated human capital (Sit and Experit) so that

�it = �1 � Sit + �2 � Experit + �3 � Exper2it + �ei (3)

where �ei is an individual speci�c intercept term, �1 represents the employmentsecurity return to schooling, both �2 and �3 represent the employment security

2When faced with a high failure probability, some individuals may spend a portion of theyear in school and a residual portion out of school. As a result, identifying a real interruptionfrom a true academic failure is tenuous. In the NLSY, we �nd that more than 85% of thesample has never experienced school interruption.

3It follows that E log et = � exp(�t+1

2�2e) and that V ar(log et) = exp(2�t+�2

e) �(exp(�2

e)�

1):

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return to experience. All random shocks ("�it; "wit; "

eit) are assumed to be indepen-

dent.In order to express the solution to the dynamic programming problem in a

compact fashion, it is convenient to summarize the state variables in a vector(St; �t) where �t is itself a vector containing the interruption status (It); theutility shock ("�t ), the wage shock ("wt ); accumulated experience (Expert) anda set of individual characteristics. As it is done often in dynamic optimizationproblems, the solution to the stochastic dynamic problem can be characterizedusing recursive methods (backward induction). The decision to remain in school,given state variables St and �t, denoted V

st (St; �t); can be expressed as

V st (St; �t) = E log(�t) + "

�t + �f� � EV I

t+1(St+1; �t+1)

+(1� �) � EMax[V st+1(St+1; �t+1); V

wt+1(St+1; �t+1)]g

or, more compactly, as

V st (St; �t) = log(�t) + �E(Vt+1 j dt = 1) (4)

where V It (St; �t) denotes the value of interrupting schooling acquisition and where

E(Vt+1 j dt = 1) denotes the value of following the optimal policy next period(either remain at school or start working). As we do not distinguish betweenincome support while in school and income support during an interruption, thevalue of entering the interruption status, V I

t+1(St; �t); can be expressed in a similarfashion.

The value of stopping school (that is entering the labor market) at the be-ginning of period t, at wage wt and with St years of schooling, while takinginto account the distribution of et (because et is unknown when wt is drawn),V wt (St; �t), is given by

V wt (St; �t) = log(wt � et) + �E(Vt+1 j dt = 0) (5)

where E(Vt+1 j dt = 0) is simply

E(Vt+1 j dt = 0) =TX

j=t+1

�j�(t+1)(� exp(�j+1

2�2e)+'1(Sj)+'2:Experj+'3:Exper

2j )

Using the terminal value as well as the distributional assumptions about thestochastic shocks, the probability of choosing a particular sequence of discretechoice can readily be expressed in closed-form.

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2.1 Unobserved Ability in School and in the Market

Ability heterogeneity has 3 dimensions: school ability (��i ); market ability a�ect-ing wages (�wi ) and market ability a�ecting employment rates (�ei ): We assumethat individual ability is the sum of a deterministic component capturing thee�ect of household human capital and a stochastic component representing id-iosyncratic unobserved ability which is orthogonal to household human capital.The unobserved ability regression function is given by the following expression;

�si = �s1�father0s educ+�s2�mother

0s educ+�s3�household income+�s4�siblings+~�si

for s = �; w and e:We assume that there areK types of individuals and that each type is endowed

with a vector (~�wk ; ~��k; ~�

ek). We set K = 6.4 The probabilities of belonging to type

k; pk; are estimated using logistic transforms

pk =exp(qk)P6j=1 exp(qj)

and with the restriction normalize q6 to 0.

2.2 Identi�cation

Identi�cation of the wage and employment returns to human capital (school-ing and experience) is straightforward with data on labor market wages. Givenknowledge of the intercept term of the wage function and the employment ratefunction, data on schooling attainments can be used to identify the utility ofattending school and, especially, the importance of heterogeneity in the interceptterm of the utility of attending school.

However, estimation of our model will require normalization. Given the ab-sence of data on the utility of attending school, it will be impossible to separatethe direct e�ects of household human capital on the utility of attending school(the Æ0s) from the e�ect of household human capital on individual school ability.As a consequence, we set ��1 = �

�2 = �

�3 = �

�4 = 0: In practice, this means that our

estimates of the e�ect of parents' background are a the sum of a direct e�ect onthe utility of attending school and an indirect e�ect capturing the transmissionof ability across generations.

2.3 The Likelihood Function

In order to implement the model empirically, we must make some additionalassumptions. First, we only model the decision to acquire schooling beyond 6

4We initially started with 4 types but we were �nally able to get up to 6 types.

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years (as virtually every individual has completed at least six years of schooling).Second, we set T (the �nite horizon) to 65 years. Finally, we set the maximumnumber of years of schooling (~t) to 22.Constructing the likelihood function is rel-atively straightforward. Using the de�nitions of dt and It; it is easy to specify alltransition probabilities needed to derive the likelihood function. These transitionprobabilities characterize the decision to leave school permanently or to continuein school. Altogether, they represent all possible destinations.

The transition probabilities that de�ne the choice between interrupting schoolpermanently (start working) and obtaining an additional year of schooling, aregiven by

Pr(dt+1 = 0 j dt = 1) = (1� �) � Pr(V wt (St) � V s

t (St)) (6)

Pr(dt+1 = 1 j dt = 1) = (1� �) � Pr(V wt (St) < V s

t (St)) (7)

Pr(It+1 = 1 j dt = 1) = � (8)

where Pr(V wt (St) � V s

t (St)) is easily evaluated using (4) and (5). Equation (6)represents the probability of exercising the right to leave school permanently int+1 (implicitly assuming It+1 = 0) while equation (7) represents the probabilityof staying in school to acquire an additional year of human capital (also implic-itly assuming It+1 = 0) : Equation (8) represents the exogenous probability ofentering the interruption state. The likelihood function is constructed from dataon the allocation of time between years spent in school (It = 0; dt = 1) and yearsduring which school was interrupted (It+1 = 1; dt = 1); and on employment his-tories (wage/unemployment) observed when schooling acquisition is terminated(until 1990).

Ignoring the individual identi�cation subscript, the construction of the likeli-hood function requires the evaluation of the following probabilities;

� the probability of having spent at most � years in school (including yearsof interruption), which can be easily derived from (6), (7) and (8).

L1 = Pr[(d0 = 1; I0); (d1 = 1; I1)::::(d� = 1; I� )]

� the probability of entering the labor market in year � + 1; at observedwage w�+1; which can be factored as the product of a normal conditionalprobability times a marginal.

L2 = Pr(d�+1 = 0; w�+1) = Pr(d�+1 = 0 j w�+1) � Pr(w�+1)

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� the density of observed wages and employment rates from � +2 until 1990.Using the fact that the random shocks a�ecting the employment processand the wage process are mutually independent and are both i:i:d., thecontribution to the likelihood for labor market histories observed from �+2until 1990 is given by

L3 = Pr (f ~w�+2g::f ~w1990g) = Pr (fw�+2 � e�+2g � ::::Prfw1990 � e1990g)

The likelihood function, for a given individual and conditional on a vector ofunobserved heterogeneity components #j =

���; �w; �e

�j, is given by Li(#j) =

L1i(#j) � L2i(#j) � L3i(#j). The unconditional contribution to the log likelihood,for individual i, is therefore given by

logLi = logK=6Xj=1

pj � Li(: j #j) (9)

where each pj represents the population proportion of type #j:

3 The Data

The sample used in the analysis is extracted from 1979 youth cohort of the TheNational Longitudinal Survey of Y outh (NLSY). The NLSY is a nationallyrepresentative sample of 12,686 Americans who were 14-21 years old as of Jan-uary 1, 1979. After the initial survey, re-interviews have been conducted in eachsubsequent year until 1996. In this paper, we restrict our sample to white maleswho were age 20 or less as of January 1, 1979. We record information on ed-ucation, wages and on employment rates for each individual from the time theindividual is age 16 up to December 31, 1990.5

The original sample contained 3,790 white males. However, we lacked infor-mation on household background variables (such as household income as of 1978and parents' education).6 The age limit and missing information regarding actualwork experience further reduced the sample to 1,710. Descriptive statistics arefound in Table 1.

Before discussing descriptive statistics, it is important to describe the con-struction of some important variables. In particular, both the schooling attain-ment variable and the experience variable deserve some discussions. First, theeducation length variable is the reported highest grade completed as of May 1of the survey year. Individuals are also asked if they are currently enrolled in

5The reason for not including information beyond 1990 is that the wage data do not appearreliable for these more recent waves.

6We lost about 17% of the sample due to missing information regarding family income andabout about 6% due to missing information regarding parents' education.

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school or not. This question allows us to identify those individuals who are stillacquiring schooling and therefore to take into account that education length isright-censored for some individuals. It also helps us to identify those individualswho have interrupted schooling. Overall, young individuals tend to acquire edu-cation without interruption. In our sample, only 306 individuals have experiencedat least one interruption (Table 1). This represents only 18% of our sample andit is along the lines of results reported in Keane and Wolpin (1997). As well, wenote that interruptions tend to be short. Almost half of the individuals (45 %)who experienced an interruption, returned to school within one year while 73%returned within 3 years.

Second, unlike many studies set in a reduced-form which use potential expe-rience (age -education- 5), we use data on actual experience. The availability ofdata on actual employment rates allows use to estimate the employment securityreturn to schooling. More details can be found in Belzil and Hansen (2001).

4 Empirical Results

In Section 4.1, we discuss the e�ects of household human capital on the utility ofattending school and on labor market outcomes. In Section 4.2, we investigatethe relative importance of household human capital and unobserved ability inexplaining di�erences in schooling attainments and wages. In Section 4.3, weevaluate the true intergenerational education correlation and its macroeconomiccounterpart; the impact of human capital based economic growth on schoolingattainments of the next generation.

4.1 Parameter estimates

To facilitate presentation of the results, we split the parameter estimates in 3tables. The e�ects of household human capital on the utility of attending schooland labor market outcomes are in Table 2A. The remaining structural parametersof the utility of attending school and the return to schooling are in Table 2B.Finally, the estimates summarizing the distribution of idiosyncratic unobservedabilities are found in Table 2C.

First, we have estimated a model where the deterministic components of wageand employment abilities (�wi ; �

ei ) are made functions of household human cap-

ital (Model 1). For illustrative purposes, we have also estimated a restrictedmodel speci�cation where household human capital does not a�ect labor marketoutcomes (Model 2).7

As the principal objective of this paper is to evaluate the relative importance ofhousehold human capital and unobserved abilities, we do not discuss all parameter

7The reader should note that all household characteristics (especially education, income andsiblings) are highly correlated.

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estimates in details. Instead, we focus on those that will enable us to answers thebasic questions raised above. An in-depth discussion of the return to schoolingand the goodness of �t for a similar model speci�cation is found in Belzil andHansen (2001).8 The respective values of the mean log likelihood (-13.7227 and-13.7399) indicate that the restricted version of the model is strongly rejected. Asa consequence, our discussion of the parameter estimates will be based primarilyon those of Model 1.

The parameter estimates for the e�ects of household human capital on theutility of attending school and labor market outcomes (found in Table 2A) indi-cate clearly that, other things equal, the utility of attending school is increasingin father's education (0.0231) and household income (0.0018). Interestingly, thee�ect of female education is negative (-0.0031) but insigni�cant. The relativelyweak e�ect of female education may be explained by the fact that more educatedfemales tend to work more in the labor market and spend less time with theirchildren. The results for siblings (-0.0157) indicate that those raised in familieswith smaller number of children tend to have a higher utility of attending school.

While the e�ects of household human capital on labor market outcomes areweaker, there is support for the hypothesis that labor market ability is correlatedwith household human capital. The positive e�ects of father's education (around0.0120) on log wages and the negative e�ect on log inverse employment rates(-0.0135) indicate that father's education increase both wages and employmentrates by more than 1%. As for the utility of attending school, the e�ect of femaleeducation on wages is found to be negative (-0.0083). However, female educa-tion has virtually no e�ect on employment rates. Household income increaseswages (0.0012) but has no signi�cant e�ect on employment rates. Finally, thenumber of siblings has a signi�cant negative e�ect on wage ability (-0.0077) andan insigni�cant negative e�ect on employment rates. Taken as a whole, thereis therefore overwhelming evidence that school and labor market abilities arestrongly correlated with household human capital.

The di�erences in the parameter estimates of the household human capitalvariables between Model 1 and Model 2 (found in column 2) indicate that ignoringthe e�ects of household characteristics on labor market outcome will lead toa serious under-estimation of the e�ect of household background variables onthe utility of attending school. This is particularly true for father's education(0.0096), household income (0.0006) and siblings (-0.0067). All these values aremuch below their equivalent estimate in Model 1. This is explained by the factthat, in the most general (unrestricted) model, household human capital raisesabsolute advantages in the labor market.

8Overall, our model is able to �t the data very well. As documented in Belzil and Hansen(2001), the estimates of the wage equation reveal that assuming constant marginal returns toschooling is a serious mistake. The high level of signi�cance of the parameter estimates for thespline functions (Table 2B) indicate that a model with constant marginal (local) returns wouldbe strongly rejected.

11

The estimates reported in Table 2C illustrate the importance of unobservedabilities. There is a relatively important variation in the individual speci�c inter-cept terms of the utility of attending school as well as in the intercept terms ofthe wage function. In Model 1, the intercept terms of the wage function rangefrom 1.1323 (type 3) to 2.1083 (type 5) while the intercept terms of the utility ofattending school range from -3.7970 (type 3) to -2.1232 (type 5). Overall, thosetypes endowed with a high school ability are also endowed with a high wageintercept. This is evidence of a positive correlation between school and marketability.9 More detailed comparisons of the e�ects of household human capital andunobserved abilities is delayed to Section 4.2.

4.2 The Relative Importance of household Human Cap-

ital and Ability in Explaining Individual Schooling

Attainments and Wages.

At this stage, two questions naturally arise. Given that predicted schooling at-tainments can be decomposed into two orthogonal components, what is the rela-tive importance of household human capital and individual unobserved abilitiesin explaining individual schooling attainments? What is the relative importanceof household human capital and individual unobserved abilities in explaining la-bor market wages? To answer these questions, we have evaluated the variationsin predicted schooling explained by both sources.

4.2.1 Schooling Attainments

The variations in predicted schooling attainments, explained by a particular vari-

able or parameter, are computed as follows:q

1N

PNi=1(PRED(Si)� E(PRED(S))2

where PRED(Si) is the predicted schooling attainment of individual i obtainedwhen we �x all attributes other than the one of interest at their respective sam-ple average and where E(PRED(S)) denotes the sample average of the predictedschooling attainments (when all variables and parameters are allowed to vary).The relative dispersion observed for both sources (such as parents' human capitaland ability heterogeneity) will provide a good indication of the relative impor-tance of each factor as predicted by the structural parameters.10

The measures of dispersion characterizing predicted schooling attainments arepresented in Table 3. Overall, there is strong evidence that father's and mother'sschooling are by far the most important household background variables. When

9For more details on the \Ability Bias" and the \Discount Rate Heterogeneity Bias", seeBelzil and hansen (2001).

10As the model is non linear, the variance of predicted schooling cannot be decomposedlinearly.

12

taken as a whole, individual variations in household human capital account for amuch larger share of total variations in schooling than does ability heterogeneity.The estimated measure of variation in individual schooling attainments imputedto idiosyncratic ability heterogeneity is equal to 0.5449 in Model 1 and representsonly 50% of the variation imputed to all household attributes (equal to 1.0977).Undoubtedly, household human capital endowments are more important thanunobserved abilities in explaining cross-sectional di�erences in schooling attain-ments.

4.2.2 Wages

While household background variables account for a larger share of cross-sectionaldi�erences in schooling than do individual abilities, it is far from obvious thatthey have a similar explanatory power on labor market wages. Both school andmarket abilities have an e�ect on wages through schooling but market ability hasalso a direct e�ect on wages through the intercept terms of the wage function. Toinvestigate this issue, we have computed a measure of variation in predicted wagesthat can be imputed to household background variables and ability heterogeneity.Without loss of generality, we used predicted entry wages. The results are incolumn 2 of Table 3.

As expected, the respective variations in explained wages due to ability het-erogeneity and household background variables are quite di�erent from thoseobserved for predicted schooling attainments. The variation in predicted wages(0.2448) explained by abilities is 3 times larger than the variation explained byall household background variables (0.0768). These results are easily explained.First, household human capital has a much larger e�ect on the utility of attend-ing school than on labor market outcomes. Second, the wage return to schoolingis found to be quite low so that individual di�erences in schooling cannot explaindi�erences in wages. Third, labor market ability appears to be the prime factorexplaining predicted wages.

4.3 What is the True Intergenerational Education Corre-

lation ?

In the reduced-form literature, it is customary to document the intergenerationaleducation correlation using OLS estimates. However, OLS estimates are likelyto be unreliable. The e�ects of household human capital on schooling attain-ments, obtained from OLS regressions, ignore dynamic dimensions and cannotdistinguish between the utility of attending school and labor market outcomes.They are also incapable of identifying the relative importance of individual abil-ity and pure stochastic shocks. As our model allows us to separate the e�ects ofhousehold human capital from unobserved abilities, it is easy to generate data onschooling attainments, letting vary all observable attributes or parameters that

13

are subject to heterogeneity, and compute partial correlations between realizedschooling attainments and household background variables. An OLS regression ofsimulated schooling attainments on household background variables will providea good estimate of the various partial correlations predicted by the structural dy-namic programming model. In turn, these can be compared to the sample partialcorrelations obtained from a reduced-form OLS regression of actual schooling onhousehold background variables.

The partial correlations measured in the data are found in the last column ofTable 4. The OLS estimate of father's schooling is around 0.2073, while theestimate of mother's education is 0.1683. The e�ect of household income is0.0154 and the e�ect of sibling is -0.1454. These results are relatively standardin the literature. As seen from Table 4, the structural dynamic programmingmodel predicts much smaller e�ects of household human capital variables thanwhat is revealed by OLS estimates obtained from actual schooling attainments.The parameter estimates for father's education (0.1724 in model 1) indicates amuch weaker correlation between individual schooling attainments and father'seducation than the sample correlation. It is approximately 17% lower than theOLS estimate found in column 2. The estimates for mother's education (0.1066) iseven smaller when compared with the correlation measured in the data (0.1683).It is 40% lower than the OLS estimate of mother's education. The e�ects ofhousehold income (0.0119 and siblings (-0.1284) predicted by the structural modelare also much weaker than those measured in the data. They are approximately30% lower (in the case of household income) and 12% lower (in the case of siblings)than their OLS counterparts in column 2. Overall, there is strong evidence thatOLS estimates of the correlation between household human capital variables andindividual schooling attainments tend to over-predict the e�ect of all variables.This seems to be particularly true for mother's education and household income.

As argued before, the intergenerational education correlation, inferred fromcross-section data, has a macroeconomic counterpart. More precisely, knowledgeof the true intergenerational education correlation can be used to measure theaverage increase in the level of schooling of the next generation explained byan exogenous increase in the level of human capital of the current generation.While the e�ect of human capital and education on growth is one of the centralquestions addressed by empirical macroeconomists, it is also important to inves-tigate how household human capital a�ects schooling attainments of the nextgeneration. Although, following Lucas (1988), more general theoretical modelshave been introduced, which involve overlapping generations and human capitaltransfers across generations, very few of them have been tested empirically.11 Inour model, the intergenerational human capital transmission mechanism is rela-tively simple. An increase in education increases household income of the currentgeneration. This increase in schooling and income will, in turn, increase the util-

11See Barro and Sala-i-Martin, 1995, for a survey.

14

ity of attending school and reduce the opportunity cost of schooling for the nextgeneration. In other words, education may be seen as a by-product of economicgrowth.

To investigate this issue, we increased exogenously the level of schooling ofboth the father and mother by one year and imputed the relevant increase inhousehold income indicated by our estimates of the returns to schooling. Then,we computed the average increase in schooling attainments of the next genera-tion. We have also performed the same experiment without changing householdincome. The results are found in Table 5. On average, increasing both parents'education by one year (with the appropriate increase in family income) will raiseschooling attainment of the next generation by 0.40 year. This increase is ex-plained mostly by an increase in education. However, there exist no well-de�nedbenchmark result which is equivalent in the macroeconomic literature devotedto economic growth. As a consequence, our result cannot be evaluated easily.It nevertheless illustrates some persisting e�ects of human capital growth andindicate that an increase in the average level of education is, at the same time,one of the consequences as well as one of the causes of economic growth.

5 Conclusion

We have estimated a structural dynamic programming model of schooling deci-sions where individual heterogeneity (observed as well as unobserved) has severaldimensions; ability in school, ability in the labor market and initial endowmentsin household human capital. The econometric speci�cation of the model is quitegeneral. The structure of the model has allowed us to investigate the relativeimportance of household human capital and individual unobserved abilities inexplaining cross sectional di�erences in schooling attainments and wages. It alsoenabled us to investigate the link between the intergenerational education cor-relation (typically measured from cross-section data) and the e�ect of humancapital based economic growth on schooling attainments of the next generation(the aggregate counterpart to the intergenerational education correlation).

Our estimates indicate that the di�erences in predicted schooling explained byhousehold human capital are generally twice as large as the di�erences explainedby unobserved abilities while the di�erences in predicted wages explained byunobserved abilities are 3 times as large as the di�erences explained by householdhuman capital. These results may be explained intuitively. First, householdhuman capital has a much larger e�ect on the utility of attending school thanon labor market outcomes. Second, the wage return to schooling is found to bequite low so that individual di�erences in schooling can hardly explain di�erencesin wages. Third, labor market ability appears to be the prime factor explainingpredicted wages.

We �nd that the true intergenerational education correlation predicted by the

15

dynamic programming model is much lower than what is revealed by standardOLS regressions of actual schooling on various household human capital variables.The true partial correlation between son and father's schooling (0.17) is around17% lower than the sample correlation. The true partial correlation between sonand mother's education (0.11) is approximately 40% lower than the sample cor-relation. The e�ects of household income (0.0119 and siblings (-0.1284) predictedby the structural model are also much weaker than the sample correlations.

Various simulations indicated that an exogenous increase of 1 year in the levelof schooling of the current generation will increase schooling attainments of thenext generation by 0.4 year. While there is no benchmark result in the growthliterature based on cross-country growth rate comparisons, our �ndings illustratethe link between the intergenerational education correlation (typically found incross-section data) and the intergenerational e�ects of human capital growth atthe aggregate level. Our �ndings also illustrate the need for economic models inwhich education is a consequence as well as a cause of economic growth.

Our results suggest interesting topics of future research. If data on parentaltransfers were available, it would be interesting to distinguish between the struc-tural (direct) e�ect of parents' education from the ability e�ect. In particular,it would be interesting to evaluate how household labor supply behavior a�ectsschooling attainments and labor market outcomes of the children. Finally, aseducation �nancing requires less parental transfers in a welfare state than in aliberal economy, it would be interesting to compare the intergenerational educa-tion correlation in countries where post-secondary schooling is heavily subsidizedto the one obtained for the US economy.

16

References

[1] Barro, Robert and Xavier Sala-i-Martin (1995) Economic Growth, New-York: McGraw Hill.

[2] Belzil, Christian and Hansen, J�orgen (2001) \Unobserved Ability and theReturn to Schooling" IZA Working paper, Concordia University.

[3] Cameron, Stephen and Heckman, James (1998) "Life Cycle Schooling andDynamic Selection Bias: Models and Evidence for Five Cohorts of AmericanMales" Journal of Political Economy, 106 (2), 262-333.

[4] Eckstein, Zvi and Kenneth Wolpin (1999) \Why youth Drop Out of HighSchool: The Impact of preferences, Opportunities and Abilities" Economet-rica, vol. 67, No. 6 (November), 1295-1339.

[5] Kane, Thomas (1994) "College Entry by Blacks since 1970: The Role ofCollege Costs, Family Background, and the Returns to Education" Journalof Political Economy, 102, 878-911.

[6] Keane, Michael P. and Wolpin, Kenneth (1997) "The Career Decisions ofYoung Men" Journal of Political Economy, 105 (3), 473-522.

[7] Lazear, Edward (1980) "Family Background and Optimal Schooling Deci-sions" Review of Economics and Statistics, 62 (1), 42-51.

[8] Lucas, Robert (1988) \On the Mechanics of Economic Development" Journalof Monetary Economics 22:3-42

[9] Magnac, Thierry and David Thesmar (1998) \Identifying Dynamic DiscreteChoice Models : An Application to School-Leaving in France" Working Pa-per, CREST, Paris, France

[10] Rust, John (1987) "Optimal Replacement of GMC Bus Engines: An Empir-ical Analysis of Harold Zurcher" Econometrica, 55 (5), 999-1033.

[11] Topel, Robert (1999) \Labor Markets and Economic Growth", Working Pa-per, University of Chicago.

17

Table 1 - Descriptive Statistics

Mean St dev. # of individualsHousehold Income/1000 36,904 27.61 1710father's educ 11.69 3.47 1710mother's educ 11.67 2.46 1710# of siblings 3.18 2.13 1710prop. raised in urban areas 0.73 - 1710prop. raised in south 0.27 - 1710prop in nuclear family 0.79 - 1710Schooling completed (1990) 12.81 2.58 1710# of interruptions 0.06 0.51 1710duration of interruptions (year) 0.43 1.39 1710wage 1979 (hour) 7.36 2.43 217wage 1980 (hour) 7.17 2.74 422wage 1981 (hour) 7.18 2.75 598wage 1982 (hour) 7.43 3.17 819wage 1983 (hour) 7.35 3.21 947wage 1984 (hour) 7.66 3.60 1071wage 1985 (hour) 8.08 3.54 1060wage 1986 (hour) 8.75 3.87 1097wage 1987 (hour) 9.64 4.44 1147wage 1988 (hour) 10.32 4.89 1215wage 1989 (hour) 10.47 4.97 1232wage 1990 (hour) 10.99 5.23 1230Experience 1990 (years) 8.05 11.55 1230

Note: Household income and hourly wages are reported in 1990 dollars.Household income is measured as of May 1978. The increasing number of wageobservations is explained by the increase in participation rates.

18

Table 2AThe E�ects of Household Human Capital

on the Utility of Attending Schooland labor Market outcomes

Model 1 Model 2

Param (st. dev) Param (st. dev)Utility ofattending School

Father's Educ 0.0231 (0.0024) 0.0091 (0.0010)Mother's Educ -0.0031 (0.0026) 0.0056 (0.0025)household Income 0.0018 (0.0003) 0.0006 (0.0003)# of Siblings -0.0157 (0.0012) -0.0067 (0.0012)

Wages

Father's Educ (�w1 ) 0.0120 (0.0021) -Mother's Educ (�w2 ) -0.0083 (0.0020) -household Income (�w3 ) 0.0012 (0.0003) -# of Siblings (�w4 ) -0.0077 (0.0025) -

Employment

Father's Educ (�e1) -0.0135 (0.0045) -Mother's Educ (�e2) 0.0026 (0.0050) -household Income (�e3) -0.0004 (0.0005) -# of Siblings (�e4) 0.0076 (0.0049) -

Note: Household income is divided by 1000.

19

Table 2BOther Structural Parameters

Model 1 Model 2Param.(Std error) Param.(Std error)

Utility in SchoolNuclear Family 0.0183 (0.0057) 0.0187 (0.0046)Rural -0.0039 (0.0048) -0.0037 (0.0043)South -0.0190 (0.0051) -0.0194 (0.0049)Stand.Dev.(��) 0.1940 (0.0105) 0.2165 (0.0116)

Splines Æ7�10 0.1035 (0.0105) 0.1058 (0.0078)Splines Æ11 0.4831 (0.0218) 0.4810 (0.0214)Splines Æ12 -1.8526 (0.0258) -1.9135 (0.0230)Splines Æ13 -1.4753 (0.0547) -1.6561 (0.0230)Splines Æ14 3.0402 (0.0118) 3.0744 (0.0132)Splines Æ15 -0.5811 (0.0234) -0.4331 (0.0141)Splines Æ16 1.0164 (0.0244) 1.1518 (0.0087)Splines Æ17�more -0.5497 (0.0084) -0.5974 (0.0235)Interruption Prob. 0.0749 (0.0036) 0.0749 (0.0036)Discount Rate 0.0030 (0.0001) 0.0032 (0.0001)Emp. Return to Schoolingschooling -0.0682 (0.0041) -0.0683 (0.0026)experience -0.0153 (0.0026) -0.0151 (0.0026)experience2 0.0001 (0.0001) 0.0001 (0.0001)std. dev (�e) 1.3473 (0.0096) 1.3461 (0.0020)Wage return to schoolingSpline grade 7-12 0.0104 (0.0002) 0.0132 (0.0003)Spline grade 13 0.0131(0.0016) 0.0153 (0.0013)Spline grade 14 0.0848 (0.0018) 0.0919 (0.0017)Spline grade 15 -0.0105 (0.0020) -0.0047 (0.0016)Spline grade 16 0.0167 (0.0022) 0.0233 (0.0019)Spline grade 17-more -0.0050 (0.0013) -0.0016 (0.0012)experience 0.0835 (0.0016) 0.0822 (0.0016)experience2 -0.0025 (0.0001) -0.0025 (0.0002)std. dev (�w) 0.2966 (0.0024) 0.2965 (0.0024)mean log Likelihood -13.7227 -13.7399

Note: The local returns computed from the spline parameter estimates are asfollows 0.0104 (7-12), 0.0235 (13), 0.1083 (14), 0.0978 (15), 0.1145(16), 0.1095 (17and more). In Model 2, the corresponding estimates are 0.0132, 0.0285, 0.1204,0.1157, 0.1390 and 0.1374.

20

Table 2CIndividual Speci�c Intercept Terms and Type Probabilities

Model 1 Model 2

Param. (St Error) Param. (St Error)

Type 1 ��1 School ab. -2.9248 (0.0108) -2.9189 (0.0090)�w1 Wage 1.4594 (0.0105) 1.4656 (0.0098)�01 Employment -3.2637 (0.0312) -3.2823 (0.0221)q1 Type Prob. 1.0146 (0.0560) 0.9634 (0.0226)

Type 2 ��2 School ab. -2.5286 (0.0125) -2.5488 (0.0088)�w2 Wage ab. 1.8687 (0.0107) 1.8739 (0.0094)�02 Employment -3.1568 (0.0384) -3.1501 (0.0077)q2 Type Prob 0.5338 (0.0518) 0.6036 (0.0075)

Type 3 ��3 School ab. -3.2681 (0.0131) -3.2560 (0.0096)�w3 Wage 1.1323 (0.0156) 1.1321 (0.0136)�03 Employment -3.0886 (0.0381) -3.1053 (0.0060)q3 Type Prob 0.0713 (0.0185) -0.0080 (0.0076)

Type 4 ��4 School ab. -3.7970 (0.0243) -3.9498 (0.0187)�w4 Wage 1.4490 (0.0162) 1.4651 (0.0070)�04 Employment -1.4903 (0.0362) -1.5014 (0.0268)q4 Type Prob -0.1636 (0.0415) -0.1847 (0.0067)

Type 5 ��5 School ab. -2.1232 (0.0289) -2.0870 (0.0185)�w5 Wage 2.1083 (0.0221) 2.1587 (0.0101)�05 Employment -3.6275 (0.0224) -3.6834 (0.0105)q5 Type Prob -1.1892 (0.0817) -1.0816 (0.0008)

Type 6 ��6 School ab. -2.6797 (0.0184) -2.6302 (0.0139)�w6 Wage 1.6612 (0.0120) 1.6909 (0.0130)�06 Employment -3.6508 (0.0173) -3.7121 (0.0157)q6 Type Prob 0.0 (normalized) 0.0 (normalized)

Note: The type probabilities are estimated using a logistic transform. Theresulting probabilities are 0.36 (type 1), 0.22 (type 2), 0.14 (type 3), 0.11 (type4), 0.04 (type 5) and 0.13 (type 6). In Model 2, the probabilities are 0.34 (type1), 0.24 (type 2), 0.13 (type 3), 0.11 (type 4), 0.04 (type 5) and 0.14 (type 6).

21

Table 3Sources of Variations in Schooling Attainments and Wages

Model 1 Model 1

Variations in Variations inpred. schooling pred. wages

Variable/Parameter

Household backgroundParents' 0.8162 0.0436education

Parents'educ 0.9843 0.0700and income

All household 1.0977 0.0768variables

School and MarketAbilities

(~�wk ; ~��k; ~�

ek) 0.5449 0.2448

All Sources1.3071 0.2762

Note: Our estimates of the variations in the predicted endogenous variable ofinterest Y (either schooling attainments or log entry wages) explained by a partic-ular source of variation (either household human capital or unobserved abilities)are computed as follows:

vuut 1

N

NXi=1

(PRED(Yi)� E(PRED(Y ))2

where PRED(Yi) is computed at the sample average of all variables or parametersother than the one of interest and where E(PRED(Y ) denotes the sample averageof the variable.

22

Table 4Estimates of the Partial Correlations between Individual Schooling

Attainments and Household Human Capital

Realized schooling Actual schooling(Model 1) (Data)

Father's educ 0.1724 (10.50) 0.2073 (10.7)

Mother's educ 0.1066 (4.51) 0.1683 (6.20)

Household Income 0.0119 (6.25) 0.0154 (7.40)

Siblings -0.1284 (5.05) -0.1454 (5.66)

R2 0.242 0.306

Note: T statistics are in parentheses.

Table 5Human Capital, Growth and Intergenerational Transfers:

The e�ect of an Exogenous Increase in Parents' Education onSchooling Attainments of the Next Generation

Model 1 Model 1Experiment .� EDUCATION OFCURRENT GENERATION 1 year 1 year

� INCOME OFCURRENT GENERATION no yes

� EDUCATION OFNEXT GENERATION 0.295 year 0.383 year

Note: Household income is increased according to the estimated re-turns to schooling

23

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